1960 AMC 12 Problem 31

Attempt Problem 31 of the 1960 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1960 AMC 12 solutions, or check the answer key.

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31.

For

x^2+2x+5

to be a factor of

x^4+px^2+q,

the values of

p

and

q

must be, respectively:

-2,

5

5,

25

10,

20

6,

25

14,

25

Answer: D

Concepts:polynomial algebraic manipulation factor

Difficulty rating: 1690

Small hint:

Work modulo

x^2+2x+5,

so

x^2=-2x-5

Big hint:

Reduce

x^4

to a linear expression and make both remainder coefficients zero

Solution:

Modulo

x^2+2x+5,

we have

x^2=-2x-5.

Then

x^3=10-x,\qquad x^4=12x+5.

Thus

\begin{gathered} x^4+px^2+q\\ {}\equiv(12-2p)x+(5-5p+q). \end{gathered}

Both coefficients vanish when

p=6

and

q=25.

Therefore, the correct answer is D.